Stochastic numerics for the gas-phase synthesis of...
Transcript of Stochastic numerics for the gas-phase synthesis of...
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Stochastic numerics for the gas-phase synthesis ofnanoparticles
Shraddha Shekar1, Alastair J. Smith1, Markus Sander1, Markus Kraft1
and Wolfgang Wagner2
1Department of Chemical Engineering and BiotechnologyUniversity of Cambridge
2Weierstrass Institute for Applied Analysis and Stochastics, Berlin
March 2011
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Outline
1 IntroductionMotivation
2 ModelType spaceParticle processesAlgorithm
3 Numerical study
4 Conclusion
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Motivation
What are nanoparticles? Why are they important?Particles sized between 1-100 nm.Both inorganic and organic nanoparticles find wide applications invarious fields.
Why model nanoparticle systems?To optimise industrial operations and to obtain products of highlyspecific properties for sensitive applications.To understand the molecular level properties that are difficult to beobserved experimentally.
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Motivation II
Salient features of the current model:Fully-coupled multidimensional stochastic population balance model.Describing various properties of nanoparticles at an unprecedentedlevel of detail.Tracking properties not only at macroscopic level but also at amolecular level.
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Applications of silica nanoparticles
Silica nanoparticles are amorphous and have Si:O = 1:2.
Their applications include:CatalysisBio-medical applicationsSupport material for functional nanoparticlesFillers/BindersOptics
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Physical system
To describe the system at a macroscopic level it is essential to understandit at a molecular level.
Precursor (TEOS) Flame reactor
P ≥ 1 atm
T ≈ 1100 - 1500 K
Silica nanoparticles
Indu
stria
l Sca
leM
olec
ular
Sca
le
Figure: Flame spray pyrolysis system
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Type space I
Each particle is represented as:
Pq = Pq(p1, . . . , pn(Pq),C).
Particle Pq consists of n(Pq) primary particles pi withi ∈ {1, . . . , n(Pq)} and q ∈ {1, . . . ,N}, where N is the total numberof particles in the system.
Si
O
O
O
OSi
O
OH
O
Si
Si
Si
Pq = Pq(p1,...,pn(Pq),C)
O Si
O
O
pi = pi(ηSi,ηO,ηOH)
HO
OH
HO
Figure: Type Space.CoMoGROUP March 2011 7 / 45
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Type space II
Each primary particle pi is represented as:
pi = pi (ηSi, ηO, ηOH)
where ηx (ηx ∈ Z with ηx ≥ 0) is the number of chemical units oftype x ∈ {Si,O,OH}.
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Type space III
C is a lower diagonal matrix of dimension n(Pq)× n(Pq) storing thecommon surface between two primary particles:
C(Pq) =
0 · · · 0 · · · 0
C21. . . 0 · · · 0
.... . . . . . · · ·
...
Ci1 · · · Cij. . .
...... · · ·
... · · ·...
.
The element Cij of matrix C has the following property:
Cij =
{0, if pi and pj are non-neighbouring ,Cij > 0, if pi and pj are neighbouring.
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Particle processes
Particles are transformed by the following processes:InceptionSurface reactionCoagulationSinteringIntra-particle reaction
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Particle processes
Particles are transformed by the following processes:InceptionSurface reactionCoagulationSinteringIntra-particle reaction
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Inception
Two molecules in gas phase collide to form a particle consisting of oneprimary.
HO
Si
OH
OH
OH
HO
Si
OH
OH
HO+
HO
Si
OH
OH
OH
HO
Si
OH
OH
HO
HO
Si
OH
O
OH Si
OH
OH
HO-2 H2O
[monomers] [primary particle]
Figure: Inception of primary particles from gas-phase monomers.
An inception event increases the number of particles in the system
molecule + molecule→ PN(p1,C),
C = 0.
Initial state of primary p1 given by:
p1 = p1(ηSi = 2, ηO = 1, ηOH = 6),
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Inception rate
Inception rate for each particle (Pq) calculated using the freemolecular kernel:
Rinc(Pq) =12K fmNA
2C 2g ,
NA is Avogadro’s constant, Cg is the gas-phase concentration of theincepting species (Si(OH)4),
K fm = 4
√πkBTmg
(d2g ),
kB is the Boltzmann constant, T is the system temperature, mg anddg are the mass and diameter respectively of the gas-phase moleculeSi(OH)4.
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Particle processes
Particles are transformed by the following processes:InceptionSurface reactionCoagulationSinteringIntra-particle reaction
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Surface reaction
Dehydration reaction between gas-phase monomer and particlesurface:
O
Si
O
O
Si
HO
O
+Si(OH)4
-H2O
O
Si
O
O
Si
O
O
Si
OH
OH
HO
Figure: Surface reaction between a particle and a gas-phase molecule.
Surface reaction transforms particle as:
Pq + molecule→ Pq(p1, ., pi′, .., pn(Pq),C
′),
p′i → pi (ηSi + 1, ηO + 1, ηOH + 2).
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Particle rounding due to surface reaction
Surface reaction also alters the common surface (C→ C′).Net common surface area of pi changes due to volume addition:
∆s(pi ) = (v(pi′)− v(pi ))
2σdp(pi )
,
where σ is the surface smoothing factor (0 ≤ σ ≤ 2).C′ is given by:
C ′ij =
{0, if pi and pj are non-neighbouring ,Cij + ∆s(pi ), if pi and pj are neighbouring.
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Surface reaction rate
Surface reaction rate calculated using equation of Arrhenius form:
Rsurf(Pq) = Asurf exp(− Ea
RT
)ηOH(Pq)NACg,
Asurf is pre-exponential factor (obtained from collision theory),Ea is activation energy,ηOH(Pq) is the total number of –OH sites on particle Pq.
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Particle processes
Particles are transformed by the following processes:InceptionSurface reactionCoagulationSinteringIntra-particle reaction
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Coagulation
Two particles collide and stick to each other:
+
PqPr Ps
Figure: Coagulation between two particles.
Coagulation of particles Pq and Pr forms new particle Ps as:
Pq + Pr → Ps(p1, ..., pn(Pq), p(n(Pq)+1), ..., pn(Pq)+n(Pr ),C).
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Coagulation II
Primary pi from particle Pq and primary pj from Pr are assumed to bein point contact.The matrix C(Ps) is calculated as:
C(Ps) =
...C(Pq) · · · Cij · · ·
......
. . . Cji . . . C(Pr )...
and has dimension n(Ps)× n(Ps), where n(Ps) = n(Pq) + n(Pr ).
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Coagulation rate
Coagulation rate between Pq and Pr calculated using transitioncoagulation kernel:
K tr(Pq,Pr ) =K sf(Pq,Pr )K fm(Pq,Pr )
K sf(Pq,Pr ) + K fm(Pq,Pr ),
where the slip-flow kernel is:
K sf(Pq,Pr ) =2kBT3µ
(1 + 1.257Kn(Pq)
dc(Pq)+
1 + 1.257Kn(Pr )
dc(Pr )
)× (dc(Pq) + dc(Pr )) ,
and the free molecular collision kernel is:
K fm(Pq,Pr ) = 2.2
√πkBT
2
(1
m(Pq)+
1m(Pr )
) 12
(dc(Pq) + dc(Pr ))2
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Particle processes
Particles are transformed by the following processes:InceptionSurface reactionCoagulationSinteringIntra-particle reaction
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Sintering
Sintering described using viscous-flow model.
pipj
pipj pk
No Sintering Partial Sintering Complete Sintering
Figure: Evolution of sintering process with time.
Sintering between pi and pj of a single particle Pq calculated on aprimary particle-level.
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Sintering level
Sintering level defined to represent degree of sintering between pi andpj :
s(pi , pj) =
Ssph(pi ,pj )Cij
− 213
1− 213
.
Ssph(pi , pj) is the surface area of a sphere with the same volume asthe two primaries.Pq conditionally changes depending on the sintering level s(pi , pj).Two types are defined depending upon a threshold (95%):
Partial sintering s(pi , pj) < 0.95Complete sintering s(pi , pj) ≥ 0.95
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Partial sintering
Surface areas of primaries are reduced by a finite amount.–OH sites at contact surface react to form Si–O–Si bonds:
SiO
O
OOH
Si
SiSi
Si
O
O
OHO
Si
SiSi
OH HO
-2H2O
OHOH
OH OHOH
HO
OH
OHOH OH
OH OH
OH
OH
OH
OH
OH
OH
OHOH OH
OH
OH
SiO
O
OO
Si
SiSi
Si
O
O
O
Si
SiSi
O
OHOH
OH OHOH
HO
OH
OHOH OH
OH OH
OH
OH
OH
OH
OH
OH
OHOH OH
OH
OH
pi
pj
pi
pjReactions at particle neck
Figure: Dehydration reaction due to sintering.
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Partial sintering II
Surface density of –OH sites assumed constant throughout sintering.The change in the internal variables of primaries pi and pj given by:
∆ηOH(pi ) = ∆ηOH(pj) = ρs(Pq)∆Cij/2,
∆ηO(pi ) = ∆ηO(pj) = −0.5×∆ηOH(pi ),
∆ηSi(pi ) = ∆ηSi(pj) = 0.
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Partial sintering III
Particle continuously transforms due to partial sintering as:
Pq(p1, . . . , pn(Pq),C)→ Pq(p1, .., p′i , p′j , .., pn(Pq),C
′)
where,p′i = pi (ηSi, ηO −∆ηO(pi ), ηOH −∆ηOH(pi )),
p′j = pj(ηSi, ηO −∆ηO(pj), ηOH −∆ηOH(pj)).
Element of C′ given by:
C ′ij = Cij −∆t
τ(pi , pj)(Cij − Ssph(pi , pj)) ,
where ∆t is a time interval.
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Complete sintering
pi and pj replaced by new primary p′′k .Particle transforms due to complete sintering as:
Pq(p1, . . . , pn(Pq),C)→ Pq(p1, .., p′′k , .., pn(Pq),C′′),
where new primary:
p′′k = p′′k (ηSi(pi ) + ηSi(pj), ηO(pi ) + ηO(pj), ηOH(pi ) + ηOH(pj)) .
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Complete sintering II
C is changed by removing columns and rows i and j and adding newcolumn and row k :
C′′ =
0 · · · 0 · · · 0...
. . ....
......
C ′′k1 · · · 0 · · · 0...
......
. . ....
C(n(Pq)−1)1 · · · C ′′(n(Pq)−1)k · · · 0
.
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Sintering rate
Sintering rate between pi and pj equivalent to rate of change of theircommon surface ∆Cij in time ∆t:
∆Cij
∆t= − 1
τ(pi , pj)(Cij − Ssph(pi , pj)),
Ssph(pi , pj) is the surface area of a sphere with the same volume asthe two primaries.
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Characteristic sintering time
Characteristic sintering time of pi and pj is:
τ(pi , pj) = As × dp(pi , pj)× exp(
Es
T
(1− dp,crit
dp(pi , pj)
)),
where dp(pi , pj) is the minimum diameter of pi and pj , andAs, Es and dp,crit are sintering parameters.
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Particle processes
Particles are transformed by the following processes:InceptionSurface reactionCoagulationSinteringIntra-particle reaction
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Intra-particle reaction
Reaction of two adjacent –OH sites on one particle:
OSi
OH
O
OSi
HO
O
O
Si
OSi
O
O
OSi
O
OSi
- H2O
Figure: Intra-particle reaction.
Intra-particle reaction transforms particle as:
Pq(p1, .., pi , .., pn(Pq),C)→ Pq(p1, .., p′i , .., pn(Pq),C),
wherep′i → pi (ηSi, ηO + 1, ηOH − 2).
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Intra-particle reaction rate
Deduce rate of intra-particle reaction from surface reaction rate andavergae sintering rate such that Si/O ratio of 1/2 is attained:
Rint(Pq) = Asurf exp(− Ea
RT
)ηOH(Pq)NACg
−ρs(Pq)
2
n(Pq)∑i ,j=1
Cij − Ssph(pi , pj)
τ(pi , pj)
,where ρs(Pq) = ηOH(Pq)/S(Pq) is the surface density of active sites.
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Algorithm
Input: State of the system Q0 at initial time t0 and final time tf .Output: State of the system Q at final time tf .t ←− t0,Q ←− Q0;while t < tf do
Calculate an exponentially distributed waiting time τ with parameter;
Rtot(Q) = Rinc(Q) + Rcoag(Q) + Rsurf(Q) + Rint(Q).
Choose a process m according to the probability
P(m) =Rm(Q)
Rtot(Q),
where Rm is the rate of the process m ∈ {inc, coag, surf, int};Perform process m;Update sintering level of all particles;Increment t ←− t + τ ;
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Numerical study
For a given property of the system ξ calculated using Nspcomputational particles and L number of independent runs, theempirical mean is:
µ(Nsp,L)1 (t) =
1L
L∑l=1
ξ(Nsp,l)(t).
The empirical variance is:
µ(Nsp,L)2 (t) =
1L
L∑l=1
ξ(Nsp,l)(t)2 − µ(Nsp,L)1 (t)
2.
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Confidence interval
The confidence interval IP within which there is a probability P offinding the true solution is then given by:
IP =[µ(Nsp,L)1 (t)− cP , µ
(Nsp,L)1 (t) + cP
].
cP = aP
õ(Nsp,L)2 (t)
L.
We use aP = 3.29 which corresponds to P = 0.999(99.9%)
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Error
The error e is:
e(Nsp,L)(t) =∣∣∣µ(Nsp,L)
1 (t)− ζ∞(t)∣∣∣ ,
ζ∞(t) is an approximation for the true solution which is obtained froma "high-precision calculation" with a very large number of particles.The average error over the entire simulation time is:
e(Nsp, L) =1M
M∑j=1
e(Nsp,L)(tj),
where the M time steps tj are equidistant.
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M0
The zeroth moment is the particle number density:
M0(t) =N(t)
Vsmpl
1.5 2 2.5 3 3.5 4 4.5 57.5
8
8.5
9
9.5
log(Nsp)
log(e
)(c
m−
3)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.810
8
109
1010
1011
1012
Time (s)
M0
(cm
−3)
M0
Nsp
=64, L=256
Nsp
=512, L=128
Nsp
=16384, L=4
High Precision Solution
Figure: Convergence of zeroth moment (Nsp × L = 65536). Solid line indicatesslope of -1.
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Volume
The average particle volume:
V (t) =1
N(t)Σ
N(t)q=1 V (Pq(t))
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
1
2
3
4
5
6
7x 10
−9
Time (s)
Volu
me
(cm
3)
1.5 2 2.5 3 3.5 4 4.5 5−18.8
−18.6
−18.4
−18.2
−18
−17.8
−17.6
log(Nsp)
log(e
)(c
m3)
Average Volume
Nsp
=64, L=256
Nsp
=512, L=128
Nsp
=16384, L=4
High Precision Solution
Figure: Convergence of average volume (Nsp × L = 65536). Solid line indicatesslope of -1.
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Collision diameter
The average collision diameter of a particle:
Dc(t) =1
N(t)Σ
N(t)q=1 dc(Pq(t))
0 0.2 0.4 0.6 0.80
10
20
30
40
50
60
70
80
Time (s)
Collis
ion
Dia
met
er(n
m)
1.5 2 2.5 3 3.5 4 4.5 5−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
log(Nsp)
log(e
)(n
m)
Average Collision Diameter
Nsp
=64, L=256
Nsp
=512, L=128
Nsp
=16384, L=4
High Precision Solution
Figure: Convergence of average average collision diameter (Nsp × L = 65536).Solid line indicates slope of -1.
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Computational time
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
10
20
30
40
50
60
70
80
Time (s)C
ollis
ion
Dia
met
er(n
m)
101
102
103
104
105
100
101
102
103
Nsp
Com
puta
tionalT
ime
(s)
Nsp
=64, L=256
Nsp
=512, L=128
Nsp
=163824, L=4
High Precision Solution
Nsp
=64
Nsp
=512
Nsp
=163824
High Precision Solution
Figure: Computational time for different Nsp.
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Computer-generated TEM
Figure: TEM images generated by projecting particles onto a plane. Experimentalvalues from Seto et al. (1995).
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Conclusion
Description of a detailed population balance model.Numerical studies performed.Demonstrated feasibility of using first-principles to model complexnanoparticle synthesis processes.
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Thank You!
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